Working Paper Cumulative Innovation, Growth and Welfare- Improving Patent Policy

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1 econstor Der Open-Access-Pubikationsserver der ZBW Leibniz-Informationszentrum Wirtschaft The Open Access Pubication Server of the ZBW Leibniz Information Centre for Economics Chor, Davin; Lai, Edwin L.-C. Working Paper Cumuative Innovation, Growth and Wefare- Improving Patent Poicy CESifo Working Paper, No Provided in Cooperation with: Ifo Institute Leibniz Institute for Economic Research at the University of Munich Suggested Citation: Chor, Davin; Lai, Edwin L.-C. (2013) : Cumuative Innovation, Growth and Wefare-Improving Patent Poicy, CESifo Working Paper, No This Version is avaiabe at: Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentiche oder kommerziee Zwecke verviefätigen, öffentich aussteen, öffentich zugängich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestet haben soten, geten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your persona and schoary purposes. You are not to copy documents for pubic or commercia purposes, to exhibit the documents pubicy, to make them pubicy avaiabe on the internet, or to distribute or otherwise use the documents in pubic. If the documents have been made avaiabe under an Open Content Licence (especiay Creative Commons Licences), you may exercise further usage rights as specified in the indicated icence. zbw Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics

2 Cumuative Innovation, Growth and Wefare-Improving Patent Poicy Davin Chor Edwin L.-C. Lai CESIFO WORKING PAPER NO CATEGORY 6: FISCAL POLICY, MACROECONOMICS AND GROWTH SEPTEMBER 2013 An eectronic version of the paper may be downoaded from the SSRN website: from the RePEc website: from the CESifo website: Twww.CESifo-group.org/wpT

3 CESifo Working Paper No Cumuative Innovation, Growth and Wefare-Improving Patent Poicy Abstract We construct a tractabe genera equiibrium mode of cumuative innovation and growth, in which new ideas stricty improve upon frontier technoogies, and productivity improvements are drawn in a stochastic manner. The presence of positive knowedge spiovers impies that the decentraized equiibrium features an aocation of abor to R&D activity that is stricty ower than the socia panner s benchmark, which suggests a roe for patent poicy. We focus on a non-infringing inventive step requirement, which stipuates the minimum improvement to the best patented technoogy that a new idea needs to make for it to be patentabe and noninfringing. We estabish that there exists a finite required inventive step that maximizes the rate of innovation, as we as a separate optima required inventive step that maximizes wefare, with the former being stricty greater than the atter. These concusions are robust to aowing for the avaiabiity of an additiona instrument in the form of patent ength poicy. JEL-Code: O310, O340, O400. Keywords: innovation, growth, patent poicy. Davin Chor Department of Economics Nationa University of Singapore 1 Arts Link, AS2 #06-02 Singapore davinchor@nus.edu.sg Edwin L.-C. Lai* Department of Economics Schoo of Business and Management Hong Kong University of Science and Technoogy, Cear Water Bay Kowoon / Hong Kong eai@ust.hk *corresponding author This draft: September 2013 We thank Costas Arkoakis, Angus Chu, Jonathan Eaton, Gino Gancia, Gene Grossman, Erzo G.J. Luttmer, Kaina Manova, Marc Meitz, Giacomo Ponzetto, Stephen Redding, Esteban Rossi-Hansberg, and Jonathan Voge for hepfu comments, as we as audiences at Penn State, the Phiadephia Fed, Princeton, UC San Diego, Stanford, the Barceona GSE Summer Forum, the SED meetings (Seou), and the Asian Meeting of the Econometric Society (Singapore). The work in this paper has been partiay supported by the Hong Kong Research Grants Counci (Genera Research Funds Project No ). Davin thanks the Internationa Economics Section at Princeton for their hospitaity. A errors are our own.

4 1 Introduction The economic anaysis of inteectua property rights (IPR) protection has focused on the efficacy of patents as an instrument for promoting research, growth and wefare. The buk of this iterature has arguaby focused on patent ength the duration of a patent s vaidity as the key poicy of interest. There is now a we-deveoped argument for the existence of an optima patent ength in a broad cass of modes where innovation expands upon the set of products (e.g., Nordhaus 1969; Tiroe 1988; Grossman and Lai 2004): An increase in the patent ength induces a higher rate of innovation by extending the duration of the innovator s monopoy power (the dynamic gains), and this is traded off at the margin against the consumer surpus that is conceded (the static osses). 1 In practice, however, patent protection is accorded in more ways than through the patent ength. Innovation often aso takes the form of productivity improvements, where new technoogies continuay buid upon od ones in a cumuative fashion, rather than engineer a radicay new product. To serve its purpose of encouraging innovation, it is therefore imperative for the IPR regime to protect patenthoders from incrementa ideas that woud compete away their profits too easiy. Toward this end, patent aws typicay incude causes that disquaify an innovator from obtaining a new patent on the basis of minor changes or trivia improvements. For exampe, the US patent code contains a non-obviousness requirement. Simiary, Artice 56 of the European Patent Convention (EPC) provides that an invention sha be considered as invoving an inventive step if, having regard to the state of the art, it is not obvious to a person skied in the art. In this paper, we deveop a mode of endogenous growth with ongoing productivity improvements, and appy it to investigate the effects of instituting an inventive step requirement to protect inteectua property in the above spirit. In Section 2, we first construct a genera equiibrium mode of growth driven by cumuative innovation, without yet introducing a binding inventive step requirement. We draw on the industry structure and Poisson arriva process for ideas from Kortum (1997) and Eaton and Kortum (2001), but adapt their setup to a specification where: (i) new ideas stricty improve upon existing frontier technoogies (as in the quaity adder modes of Grossman and Hepman 1991, and Aghion and Howitt 1992); and (ii) the size of the productivity improvement that each new idea makes is an independent draw from an underying Pareto distribution. We focus on a steady state in which a constant share of abor is engaged in innovation, with the remaining workers aocated to production activity. (This steady state is one to which the economy immediatey jumps foowing any shock.) Of note, the steady state sustains a positive rate of innovation and hence growth, even with a constant workforce size. The decentraized equiibrium moreover features a stricty smaer aocation of abor to R&D than that which a socia panner woud 1 Bodrin and Levine (2008) have argued that the patent system as currenty conceived and impemented, cedes too much power to incumbent patent-hoders to the extent that it has discouraged innovation effort instead. 1

5 optimay choose. We argue that this wedge arises soey from the fact that individua researchers do not internaize the positive knowedge spiovers of their R&D effort on future innovators when ideas buid cumuativey on each other. The above resut that the decentraized equiibrium features under-investment in R&D suggests that there may be room for poicies to improve upon wefare outcomes. Given the innovation process in our mode, a natura poicy to consider is an inventive step requirement, that stipuates how much of an improvement a new idea needs to make over the existing best patented technoogy to be deemed sufficienty non-obvious to quaify for a new patent. Specificay, if z is the productivity associated with the current best patent for a given product, a new idea woud need to deiver a productivity of at east Bz for it to be patentabe, where B 1 is a poicy parameter set by the patent authority. For simpicity, we wi further assume that a ideas that are patentabe wi aso be deemed to be non-infringing on the scope of a other existing patents (and vice versa), so that goods made using the new idea can be marketed without fear of ega action from incumbent patent-hoders. 2 We wi therefore refer more precisey to this poicy instrument, B, as a non-infringing inventive step (NIS) requirement. 3 We buid this patent instrument into our growth mode in Section 3 and unpack its various effects. First, a higher B raises the profits of patent-hoders as it aows them to exercise their monopoy power for a onger expected duration, by protecting them against future innovations that are too incrementa ( profit effect). At the same time, a more stringent NIS requirement raises the bar that ideas have to cear to quaify for a new patent ( hurde effect), this being the key additiona compication reative to anayses of patent ength poicy. These two effects ceary exert forces in opposite directions on the incentives to undertake research. We find that as ong as the capacity of the economy to generate ideas is sufficienty arge, the profit effect wi dominate when B is sma, so that research incentives improve when the NIS requirement is initiay raised above 1. However, the hurde effect necessariy dominates when B is arge, as an excessivey high bar woud instead discourage research. The reationship between equiibrium R&D effort and the inventive step parameter thus takes on an inverted U-shape, and there is a unique vaue of B (denoted by B v ) that maximizes the innovation rate. We then turn to the impications of this NIS requirement for the representative consumer. From a wefare perspective, the margina socia cost of setting a higher B is that it hurts consumer surpus by conferring more monopoy power to patent-hoders, this being the famiiar static oss arising from patent 2 In the patenting iterature, this is otherwise known as the eading breadth, namey the extent to which a new innovation needs to improve upon an existing patent to be considered non-infringing on the atter s patent rights. If the new invention is deemed to be infringing, the innovator woud need to pay a royaty to the incumbent patent-hoder in order to egay market the new product. The concepts of the patentabiity requirement (to quaify for a new patent) and the eading breadth (to avoid infringement) are cosey reated but distinct. See O Donoghue (1998) and Scotchmer (2004), Chapter 3, for a review of these issues. 3 In the context of this paper, the terms NIS requirement, inventive step requirement, patentabiity requirement, and eading breadth can therefore be used interchangeaby. 2

6 protection. Nevertheess, a higher NIS requirement raises innovation when B is sufficienty sma, which transates into faster reductions in prices for consumers (the dynamic gain). Our key resut estabishes that there is indeed a unique wefare-maximizing inventive step requirement (denoted by B w ), which is in genera binding (i.e., stricty bigger than 1). This optima tradeoff between dynamic gains and static osses occurs at a vaue of B where research effort is sti increasing in B, impying that the NIS requirement that maximizes the innovation rate is necessariy arger than that which maximizes wefare (B w < B v ). We extend our mode in Section 4 to incorporate patent ength poicy, given its common use in practice. When both patent ength and NIS instruments are avaiabe to the patent authority, we find that the wefare-maximizing poicy entais setting an infinite patent ength in tandem with a finite required inventive step of B = B w, this being the same B w derived earier in Section 3. This infinite patent ength resut arises because there are no diminishing returns to innovation effort in our mode the Poisson arriva rate of ideas is proportiona to the aggregate number of research workers at each date which provides strong incentives to raise the patent ength to promote innovation. These dynamic gains from extending the patent ength are aways arger than the static consumer surpus osses, so ong as the parameters that govern the innovative capacity of the economy are arge enough to begin with to ensure a positive amount of R&D activity in the steady state. In contrast, the effect of the NIS requirement on the rate of innovation is dampened by the hurde effect, which gets stronger as B increases, hence ensuring that B w is finite. We provide further discussion in this section of how the optima NIS requirement woud respond to the patent ength, if the atter were nevertheess set at a finite vaue for exogenous reasons. In terms of its structure, our framework fas within the cass of endogenous growth modes in which innovation occurs aong quaity (or productivity) adders (Grossman and Hepman 1991; Aghion and Howitt 1992). Firms compete by investing in R&D and successfu innovation aows them to cimb onto the next rung of the adder, a process resembing the patent race in Reinganum (1985). We however augment our mode with features drawn from Kortum (1997) and Eaton and Kortum (2001), so that the size of each innovation step is stochastic rather than deterministic. Reative to Kortum (1997), our approach specifies productivity improvements, rather than productivity eves, to be the stochastic outcomes of R&D effort, so that each new idea buids on its predecessor in a stricty cumuative fashion and steady-state growth emerges endogenousy. 4 Our motivation for modeing this truy cumuative innovation process stems in turn from our interest in exporing the scope for patent poicy intervention in the presence of knowedge spiovers, which are absent in the baseine mode of Kortum (1997). We shoud moreover stress that the mode we deveop is very tractabe for anayzing the research 4 This differs from the approach in severa recent contributions where steady state growth arises instead through the earning or diffusion of ideas from high to ow productivity firms; see for exampe, Avarez et a. (2008), Lucas and Mo (2012), Luttmer (2007), Pera and Tonetti (2012). 3

7 questions at hand. When each productivity improvement is independenty drawn from an underying Pareto distribution, we show that the og productivity eve of the best idea inherits a Gamma distribution, an observation that faciitates expicit expressions for wefare and the growth rate. Convenienty, a steady-state outcomes in the mode end up depending ony on three deep parameters, namey the rate of time preference, the Pareto dispersion parameter, and what we sha term the innovative capacity of the economy. This parsimony and tractabiity aows us to perform a cear decomposition of the effects of patent poicy on the innovation rate and wefare in a genera equiibrium setting. Whie severa other papers have aso worked with this specification of Pareto productivity improvements in an endogenous growth setting (Koéda 2004; Minniti et a. 2011), our approach differs from this prior work in the extent of the cosed-form anaytics that we pursue. On a separate note, our paper is naturay connected to work in industria organization studying the effects of instruments reated to the non-obviousness criterion in the patent code. In particuar, O Donoghue (1998) concudes that a patentabiity requirement can ead to an improvement in socia wefare, whie Hunt (2004) finds an inverted-u shape reationship between the rate of innovation and the strength of this requirement. 5 Whie these echo severa of our findings, most of the resuts in this prior work are derived in a partia equiibrium setting. An exception to this is O Donoghue and Zweimüer (2004), who embed such patentabiity considerations into a quaity-adder endogenous growth mode and derive conditions under which such poicies can raise innovation, athough their approach does not appear to deiver a cean wefare anaysis. 6 The question of the optima combination of patent ength and patent breadth poicies has aso been expored in this iterature (see for exampe, Gibert and Shapiro 1990, Kemperer 1990, Gaini 1992, Denicoò 1996), athough as we sha make cear in Section 4, the concept of the patent breadth differs from the NIS instrument that we consider. The non-monotonic reationship that we find between the strength of the NIS requirement and innovation outcomes bears paraes with severa papers (incuding Hunt (2004), mentioned above). It has been observed that increased patent protection does not necessariy transate into a faster rate of innovation; see for exampe Sakakibara and Branstetter (2001) on Japan, Bessen and Maskin (2009) on the US software industry, and the extensive review in Bodrin and Levine (2008, Chapter 8). These patterns can be rationaized within our mode if the NIS requirement has been set too high, namey at a eve of B above B v. In this range, the hurde effect dominates and owers the ex-ante chance of an idea being patentabe, eading to ess innovation when B is raised. On this note, Bessen and Maskin (2009) provide 5 Green and Scotchmer (1995) tacke a simiar probem, but focus on the eading breadth (the minimum required inventive step to avoid infringement of existing patents), as opposed to the patentabiity requirement, as their poicy instrument of interest. O Donoghue et a. (1998) consider both eading and agging breadth poicies, where the atter serves to protect patent-hoders against imitators. 6 On a reated note, Li (2001) anayzes the effect of agging patent breadth (protection against imitators) in a quaityadder growth setting. Kwan and Lai (2003) incorporate patent ength considerations into an endogenous growth mode in the vein of Romer (1990). 4

8 a reated but distinct expanation, namey that the rate of innovation can decine in the strength of patent protection when innovations are sequentia and different ines of research effort compement each other. More broady, the above body of work speaks to the issue of the appropriate eve of protection new entrants shoud receive against an incumbent s anti-competitive behavior in order to induce faster innovation and/or raise wefare. Sega and Whinston (2007) provide a recent contribution here in the context of antitrust poicy; of note, they recognize that the eve of antitrust protection can have both positive and negative effects on the rate of innovation, when the atter is ongoing and cumuative. The rest of the paper proceeds as foows. Section 2 deveops our baseine mode of cumuative innovation and anayzes the wefare properties of the decentraized equiibrium. We incorporate a binding NIS requirement in Section 3 and derive our key resuts on the scope for this patent instrument to raise innovation and wefare. We extend the mode to an anaysis of patent ength poicy in Section 4. Section 5 concudes. Detaied proofs are presented in the Appendix. 2 Baseine Mode with No Inventive Step Requirement We first buid and sove our baseine mode of cumuative innovation and growth, in order to famiiarize readers with the key features of the setup. Accordingy, in this section, innovators wi not face a binding inventive step requirement for their ideas to be both patentabe and marketabe (non-infringing). For now, we sha aso assume that patents do not expire, namey that the patent ength is infinite, so that incumbent patent-hoders ose their monopoy power ony when superseded by a new innovation. This wi pace the focus on the innovation process in our mode, which we proceed to describe next. We wi then cose the mode in genera equiibrium and discuss its properties. 2.1 Mode setup: The innovation process Consider an economy composed of one industry, in which a continuum of differentiated varieties indexed by j [0, 1] is produced. 7 The economy is endowed with L units of abor, which is the ony factor of production. A of this abor is ineasticay suppied at the wage, w τ, where τ indexes time. Firms in the economy are sma, in the sense that each firm produces ony one variety, whie taking the prevaiing wage as given. (In the aggregate, however, w τ wi be an outcome of the genera equiibrium of the mode, as we wi see beow.) Each unit of abor (or simpy worker ) can be engaged in one of two activities, namey either in the production of differentiated varieties or in R&D activity. With regard to the former, production takes pace under a constant returns-to-scae technoogy. Let Z τ (j) denote the abor productivity associated 7 It is straightforward to extend the mode to incude a non-innovating outside sector, whose output can then pay the roe of the numeraire. 5

9 with the best avaiabe idea for producing variety j at time τ; 1/Z τ (j) units of abor are thus required to produce each unit of this variety. Then, the unit cost faced by the firm that produces this variety is simpy: w τ /Z τ (j). On the other hand, the objective of R&D activity is to generate ideas to improve upon existing technoogies. Each idea spes out a technoogy (equivaenty, a abor productivity eve) for a specific differentiated variety. We mode the generation of these ideas as a Poisson process with a constant arriva rate of λ for each R&D worker. 8 Foowing Kortum (1997) and Kette and Kortum (2004), conditiona on receiving a new idea, the identity of the variety to which the idea appies is determined by a random draw from a uniform distribution on the unit interva. 9 We specify a setting in which the innovation process is stricty cumuative. In particuar, knowedge about production technoogies diffuses immediatey as soon as the good in question is marketed, so that the underying knowhow becomes avaiabe to a agents in the economy. For exampe, this coud be because it is easy to reverse-engineer the technoogy after observing a physica sampe of the good. As the current best technoogy for producing each marketed good is widey-known, subsequent innovation effort stricty buids upon this knowedge to generate productivity improvements. In equiibrium, the best patented technoogy for each differentiated variety wi indeed be used in production, with the good being marketed, and hence each subsequent arriving idea wi aways improve upon the frontier patented technoogy for the variety to which it appies. 10 In the absence of IPR protection, the diffuse nature of knowedge woud provide itte incentive for private agents to undertake R&D. We therefore require that an IPR regime be in pace that aows any new idea to be patented at negigibe cost. By patenting a new idea, the firm in possession of that idea gains excusive rights to produce and market the variety (say, variety j) with the new technoogy, and wi indeed have the entire market for j to itsef as it is now the most productive manufacturer of j. This monopoy power ony expires when the next idea that improves upon the technoogy for j arrives. 11 Ideas that are not patented but which are marketed can immediatey be egay imitated by other firms, which woud compete away the profits accruing to the origina innovator. It foows that firms wi immediatey patent any new ideas that they receive, so that no goods wi be marketed without first being patented. Having spet out the arriva process for ideas, we now describe what governs the productivity eves associated with these ideas. To initiaize the innovation process, we assume that at the start of time 8 In other words, the probabiity that an individua worker wi receive a new idea during a sma time interva τ is given by λ τ. Moreover, each R&D worker can receive ony one idea at any instant in time. 9 As in these preceding papers, this rues out the possibiity that innovation effort can be directed toward the production of specific varieties. 10 Aternativey, the innovation process can be set up as one entaiing improvements aong a quaity dimension, where each arriving idea yieds a higher utiity to consumers with no change in the good s production cost (and hence market price). 11 Given the continuous measure of varieties, there is a zero probabiity that the same agent wi consecutivey receive two ideas for producing the same variety. 6

10 (τ = 0), there is a baseine technoogy for each variety that is freey avaiabe to a firms. We normaize the productivity of this baseine technoogy to be 1 for a varieties, namey: Z 0 (j) = 1 for a j [0, 1]. Now, define Z (k) (j) to be the productivity associated with the k-th idea to arrive (after time 0) for variety j, where k is a non-negative integer. Thus, {Z (0) (j), Z (1) (j), Z (2) (j),...} form a sequence of the successive best technoogies for producing this variety. To describe how this frontier technoogy evoves, define ζ (k+1) (j) Z (k+1) (j)/z (k) (j) to be the productivity improvement associated with the next idea to arrive. We specify ζ (k+1) (j) to be a random variabe that is an independent draw from the foowing standardized Pareto distribution with shape parameter θ > 1: ( ) Pr ζ (k+1) (j) < z = 1 z θ, where z [1, ), for a k 0. (1) Note that a ower θ impies a more fat-taied distribution which paces greater weight on drawing reativey arge productivity improvements. For simpicity, the distribution in (1) does not depend on j, so that the underying innovation process is symmetric across varieties. Moving forward, we wi thus write Z (k) (j) simpy as Z (k), since the distribution of the productivity eve of the k-th idea to arrive wi be identica for a varieties. 12 Observe first that (1) embodies the notion of cumuative innovation, since the ower bound of the support of the distribution of productivity improvements is 1. In effect, after the k-th idea has arrived, the productivity Z (k) associated with that idea becomes the new knowedge frontier which the (k + 1)- th idea wi improve upon. This is consistent with our setting in which knowedge of a marketed idea immediatey diffuses through the whoe economy, and a subsequent innovation effort then buids upon it. Note aso that we have assumed that the distribution in (1) does not depend on how many ideas have aready arrived (k) or on the productivity eve of the ast drawn idea (Z (k) ). In sum, this means that conditiona on the reaized vaue of Z (k), the next arriving idea Z (k+1) can be viewed as a draw from a Pareto distribution with the same shape parameter but with a ower bound of Z (k). 13 At this juncture, it is usefu to discuss the reationship between the innovation process that we have just described and that advanced in Kortum (1997) and Eaton and Kortum (2001). In the notation that we have adopted, the anaogue of their specification for the (stationary) distribution that governs innovation is: Pr(Z (k+1) < z) = 1 z θ, where z [1, ), for a k 0. (2) Thus, in this earier work, ideas that arrive may or may not surpass the current state-of-the-art technoogy, Z (k) ; those ideas that fa short of the frontier are not competitive enough to survive in the market. As 12 This Pareto specification for each productivity improvement is aso adopted by Koéda (2004), Minniti et a. (2011), and Desmet and Rossi-Hansberg (2012). In particuar, Minniti et a. (2011) provide descriptive evidence of: (i) substantia cross-firm heterogeneity in the usefuness of innovations (as captured by patent citations), and (ii) the Pareto providing a reasonabe fit to the distribution of the vaue of patents especiay in its right-tai. 13 Reca that if a Pareto distribution is truncated from the eft, the resuting distribution remains Pareto with the same shape parameter, but with the eft truncation vaue serving as the new ower bound of its support. 7

11 more ideas accumuate over time in their economy, it becomes ess ikey that a new idea wi surpass the current frontier. In contrast, our interest ies in understanding an innovation process in which each new idea stricty improves upon Z (k). The two approaches therefore represent two opposite ends of the spectrum: Whie Kortum (1997) and Eaton and Kortum (2001) adopt a non-cumuative formuation, we instead expore a situation where innovation is fuy cumuative, this being motivated by our interest in anayzing the externaities that arise from R&D activity in this atter setting. 2.2 Genera equiibrium We now embed the above cumuative innovation process in a genera equiibrium setting. Utiity: The utiity function of the representative consumer as of date 0 is given by: U 0 = 0 e ρτ n u τ dτ. (3) Here, ρ is the rate of time preference (a parameter), whie u τ aggregates the instantaneous utiity from the consumption of differentiated varieties at time τ. Specificay, u τ is given by: { 1 } u τ = exp n x τ (j) dj, (4) 0 where x τ (j) denotes the quantity of variety j consumed at time τ. The representative consumer chooses {x τ (j)} τ=0 in order to maximize (3), subject to the intertempora budget constraint: 0 e rτ X τ dτ b(0), (5) where X τ = 1 0 p τ (j) x τ (j) dj denotes the fow of consumption spending at time τ, with p τ (j) being the corresponding price of variety j at that time (which the consumer takes as given). r is the prevaiing interest rate, which wi be pinned down by the rate of return earned on owning a patent, this being the ony asset in our economy. Finay, b(0) denotes the present vaue of the future stream of wage income that wi be earned by each consumer pus the vaue of her initia asset hodings at date 0. It is we-known (see for exampe, Grossman and Hepman, 1991) that the soution to this dynamic optimization probem yieds: r = ρ + Ẋ X, (6) where Ẋ is the time derivative of X. (We wi omit the time subscript for equations that hod for a τ 0.) Thus, the rate of growth of consumption spending shoud equa the difference between the market interest rate and one s private rate of time preference. It wi now be convenient to set aggregate consumption expenditure E τ LX τ as the numeraire for each τ. Since E τ = 1 and L is constant over time, it foows from (6) that r = ρ. Moreover, the expenditure on each variety wi be constant and equa 8

12 to 1 at each date, since we have a unit measure of varieties. As the expenditure on each j is invariant to its price, the price easticity of demand for each variety is 1. Market structure and profits: Firms compete by setting prices. If no ideas have yet arrived for variety j by time τ, then that variety is priced at margina cost (w τ ), since the baseine technoogy is freey accessibe to a potentia producers. On the other hand, if at east one idea has arrived for variety j, then the market structure becomes one of Bertrand competition. The firm possessing the most productive idea wi set a imit price that is just enough to keep the second most productive firm (and by impication, a other firms) out of the market. Therefore, the equiibrium price for variety j at time τ when precisey k ideas have arrived is equa to: p τ (j) = w τ /Z k 1 (j). The price markup that the most productive firm sets is simpy: Z k (j)/z k 1 (j), so that it inherits the Pareto distribution from (1). To be more expicit, denoting µ(m) as the cdf of the price markup m, we have: µ(m) = 1 m θ for m 1. The expected fow profits earned by a firm which hods the patent for the best technoogy for a given variety can now be computed as: ( ) m 1 Π τ = E τ dµ(m) = 1 m 1 + θ. (7) m=1 The above makes use of the fact that E τ is aso the expenditure per variety, since we have a unit measure of differentiated varieties. 14 Savings and investment: Let v [0, 1] denote the share of the abor endowment L that is hired by firms to engage in R&D activity. (The remaining fraction, 1 v, works in the production of goods.) We assume that firms need to obtain financing in order to hire R&D workers. This spending on R&D is the ony form of investment in our mode, in the sense that the innovation is undertaken to generate ideas that yied a future stream of profits. In exchange for the financing they obtain, firms issue caims on the fow of profits from their patents. These caims (which we can think of as equity) are the ony assets in the economy, and the tota vaue of these assets at time τ is denoted by A τ. (Note that A τ is aso equa to the vaue of each patent, given that we have a unit measure of varieties and ony one active patent for each variety.) The above investment in R&D activity is financed through the savings of workers. This coud take pace with savers directy owning the equity of firms, or with the financing channeed through an intermediary such as a bank. Figure 1 summarizes this circuar fow of funds in the economy between workers, firms and the financia intermediary sector (or capita market). Bear in mind that workers have two sources of income, namey their abor income and the return that they earn on assets. The aggregate savings in the economy are thus equa to tota income net of consumption spending, wl + ra E. Equating this with aggregate net investment ( A) at each point in 14 To be cear, Π is equa to profits for a variety conditiona on at east one idea having arrived for the variety in question. In particuar, Π is not equa to aggregate profits in the economy. 9

13 Househods / Workers vwl Capita Market E wl vwl Firms Figure 1: Circuar Fow in the Economy time, we have: wl + ra 1 = A. (8) Research incentives: Since vl workers are empoyed in R&D, the Poisson arriva rate for new ideas in the economy as a whoe is equa to λvl. The rate of return r for owning an asset (namey, a patent) must equa the fow profit rate minus the probabiity of a compete capita oss due to the arriva of a new idea that supersedes the existing patent. This impies: r = Π λvl. (9) A As r = ρ, one can rewrite (9) as A = Π/ (ρ + λvl), which gives us an expression for the expected present discounted vaue of each patent. Intuitivey, this is equa to the present vaue of fow profits, discounted by the rate of time preference pus the hazard rate of osing the market to a subsequent innovator. Labor market equiibrium: We consider an equiibrium in which a positive amount of production (and hence consumption) takes pace in each time period. This impies that the wage of a production worker needs to weaky exceed the vaue of the margina product of being an R&D worker. The atter is given by the fow rate of ideas that each R&D worker can generate mutipied by the vaue of each idea, namey λa. We thus have: λa w. (10) Equation (10) wi hod with equaity when some innovation activity takes pace, namey when v ies in the interior of [0, 1]. Workers woud then be indifferent between being empoyed in R&D and production. Steady state: The five equations (6), (7), (8), (9) and (10) define a system in the five unknowns Π, A, w, r, and v, which pins down the steady state of our mode. 10

14 In what foows, we focus on a steady state in which some innovation occurs, and in which the share of the abor force empoyed in R&D is constant over time. In particuar, this means that (10) wi hod as an equaity. A quick inspection of our system of equations then impies that the vaue of a patent (A), the return on assets (r), and the return to abor (w) wi a be constant in this steady state. Moreover, a famiiar set of arguments can be appied to show that this steady state is one to which the economy immediatey jumps. To see this, (8) and (10) together impy that: Ȧ/A = λl + ρ (1/A). If λl + ρ were to exceed 1/A at any time aong the transition path, A/A woud be positive, and the subsequent increase in A woud further widen the gap between λl + ρ and 1/A on the right-hand side. Aso, the arger is A, the faster is the rate of increase in A, so that the vaue of a patent woud continue to increase to infinity. However, (9) paces an upper bound of Π/ρ on the vaue of a patent, so that the expectation that A wi increase indefinitey cannot be met. A simiar argument can be used to rue out the reverse case where λl + ρ fas short of 1/A on the transition path. Thus, expectations about the vaue of a patent can ony be fufied if the economy jumps immediatey to a situation where A/A = 0. It is now straightforward to sove the system of five equations after setting Ẋ = A = 0. This yieds in particuar the foowing expression for the market aocation of abor to R&D activities, v eqm : v eqm = λl ρθ λl (1 + θ) (11) Note from the above that v eqm is ceary ess than 1. To further ensure that v eqm > 0, we need to impose the foowing: Assumption 1: λl > ρθ. Intuitivey, for there to be a positive amount of R&D in the steady state, we require that: (i) the innovative capacity of the economy (captured by λl) be sufficienty high; (ii) the dispersion of the ideas distribution be arge (θ sma); and/or (iii) consumers be sufficienty patient (ρ ow). Using (11), one can verify directy that the research intensity of the economy varies naturay with the underying parameters of the mode. Firms hire a greater share of the workforce in R&D when the arriva rate of ideas is higher (dv eqm /dλ > 0), or when those ideas are drawn from a Pareto distribution with a fatter right-tai (dv eqm /dθ < 0). Moreover, if agents are more patient when vauing future reative to current consumption, this aso raises R&D effort in the steady state (dv eqm /dρ < 0) Stricty speaking, our mode exhibits a scae effect in that equiibrium R&D effort is increasing in L. We shoud stress nevertheess that a economic outcomes of interest in our mode, such as v, growth rates and wefare, ony depend on the product λl, and not on the specific vaues of λ and L separatey. What is more important is therefore not the size of the economy as measured by L, but its innovative capacity as captured by λl. 11

15 2.3 Wefare We turn next to the task of evauating country wefare, in order to faciitate our ater anaysis of the efficacy of patent poicy. The utiity specification in (3) and (4) impies that wefare depends on the rea { } 1 wage in each period, as: u τ = w τ / exp 0 n p τ (j)dj. (Reca in particuar that the economy jumps immediatey to its steady state.) Since a varieties are ex ante symmetric and we have a unit measure of these varieties, the aw of arge numbers impies that the idea price index in the denominator is equa to: exp {E[n P τ ]}, where P τ is a random variabe whose reaization is the price of a variety at time τ; the expectation operator is taken over this price distribution. We therefore need to understand how prices evove over time. Due to the Poisson nature of the innovation process, the probabiity that exacty k ideas have arrived by time τ when vl units of abor are engaged in R&D at each date is: (λvlτ)k k! e λvlτ, where k is a non-negative integer. Reca that when k = 0, the variety in question wi be priced at w τ (its margina cost). On the other hand, when k 1, under the imit-pricing rue, the price of a variety wi instead be a random variabe that inherits the distribution of w τ /Z (k 1). The expected og price of a variety at time τ is thus: E [n P τ ] = (λvlτ)0 e λvlτ n w τ + 0! (λvlτ) k k=1 k! ( ) e λvlτ n w τ E[n Z (k 1) ]. (12) Note that the first term and each term in the summation in (12) is equa to the probabiity that k ideas have arrived between times 0 and τ, mutipied by the og price at time τ when there have indeed been exacty k ideas (where k = 0, 1, 2,..., ). We show in the Appendix how to evauate (12) expicity. The key to this is to recognize that in the underying innovation process, the random variabe Z (k 1) = Z (k 1) /Z (0) is the product of k 1 independent reaizations from the standardized Pareto distribution given earier in (1). (Reca that Z (0) = 1.) Buiding off this observation, one can show that n Z (k 1) is a random variabe from a Gamma distribution with mean E[n Z (k 1) ] = (k 1)/θ (see the Appendix). 16 The expected og productivity of the k-th idea to arrive thus increases ineary in k, whie increasing aso in the thickness of the right-tai of the Pareto distribution from which the productivity improvements are drawn. Substituting this expression for E[n Z (k 1) ] into (12) and simpifying, one then obtains: E [n P τ ] = n w τ + 1 ( θ 1 λvlτ e λvlτ ). 17 It foows that per-period utiity (the rea wage) is given by: u τ = exp { 1 ( θ 1 λvlτ e λvlτ )} To be absoutey precise, this statement about the distribution of n Z (k 1) hods ony for k 2. Nevertheess, when k = 1, we have that E[n Z (0) ] = 0, so that the formua E[n Z (k 1) ] = (k 1)/θ is aso vaid for k = Much work has been done documenting the fit of the Pareto distribution for firm size distributions (e.g., Axte 2001; Luttmer 2007; Arkoakis 2011). Interestingy, the Gamma distribution aso features a thick right-tai, athough it matches the empirica distribution of US firms ess we for the argest firm sizes (Luttmer 2007). 18 The nomina wage can aso be soved for expicity from the system of five equations that pin down the steady state. This is given by: w τ = λ/(ρ + λl). 12

16 Defining the growth rate of the rea wage to be g τ d n (u τ ) /dτ, we have: g τ = λvl θ ( 1 e λvlτ ), (13) which is ceary positive when v > 0. Athough the economy jumps immediatey to a steady state in which A, Π, and w (the nomina wage) are constant, the rea wage nevertheess rises over time as varieties are on average becoming cheaper when there is a positive amount of R&D. In other words, Assumption 1 which guarantees that v eqm > 0 aso ensures that g τ > 0 for a τ 0. Substituting in the expression for v eqm from (11), one can further verify that: dg τ /dλ > 0 and dg τ /dθ < 0. Thus, a higher arriva rate of ideas (higher λ) and a arger average productivity improvement (smaer θ) both raise the growth rate of the rea wage at each date τ. From (13), one can moreover see that the growth rate of the rea wage rises over time (dg τ /dτ > 0): From an initia vaue of g τ = 0, this asymptotes toward a maximum growth rate of λvl/θ. This property derives from the fact that as time progresses, the baseine technoogy is shed from use for a greater and greater share of varieties. As the first idea arrives for successive varieties, the innovation process gets jump-started for a greater measure of varieties in the unit interva, hence causing the overa growth rate to rise over time. However, this effect peters out, as the first idea eventuay arrives in expectation for a varieties. It is instructive here to compare the above against the properties of the modes in Kortum (1997) and Eaton and Kortum (2001), which aso focus on a steady state in which the share of the workforce empoyed in R&D is constant. In these preceding papers, innovation is not cumuative in nature, and perpetua growth in rea wages is sustained instead by a growing R&D workforce (vl), which grows at the same exogenous rate as the abor force (L). Thus, more ideas are drawn in each period by the ever-growing number of R&D workers, overcoming the fact that it gets harder and harder for each idea drawn from the stationary distribution in (2) to surpass the technoogica frontier. In contrast, the mode which we have just presented generates steady-state growth in rea wages through the cumuative nature of innovation new ideas aways stricty improve on the technoogica frontier without requiring that the abor force grow over time. Finay, the expected wefare of the representative consumer is obtained by substituting the expression for u τ into (3) and evauating the associated integra. After some agebraic simpification, this yieds: U 0 = (λvl) 2 ρ 2 θ(ρ + λvl) = (λl ρθ) 2 ρ 2 θ(1 + θ)(ρ + λl), (14) where the ast equaity foows from repacing v by the expression for v eqm from (11). One can show via straightforward differentiation that so ong as Assumption 1 hods, (14) is increasing in λ and decreasing in θ. Wefare therefore rises either as innovations arrive more frequenty or as the average productivity improvement increases. 13

17 2.4 Contrast with the socia optimum To understand the efficiency properties of the steady state which we have just soved for, it is instructive to compare the above market equiibrium with the outcomes under a benign socia panner. Conceptuay, this socia panner s probem can be formuated as a abor aocation decision over the share of abor to empoy in R&D, as we as the vaue of L p τ (j) for each j [0, 1], namey the amount of abor assigned to the production of variety j at each point in time. Formay, the socia panner sets out to sove: max v,{l p τ (j)} 1 j=0 s.t. 1 0 U 0 L p τ (j) dj = L(1 v) for a τ 0, (15) and Lx τ (j) = L p τ (j)z τ (j) for a j [0, 1] and τ 0. (16) The first constraint (15) is a abor market-cearing condition that states that a abor not engaged in R&D must be empoyed in production. On the other hand, the second constraint (16) sets the quantity demanded of variety j equa to the quantity produced at each period in time. As we show in the Appendix, the soution to this socia panner s probem features an equa aocation of abor to the production of each variety. In other words, given the choice of v, we have: L p τ (j) = L(1 v). Using this property, the panner s probem can then be simpified to the foowing unconstrained maximization probem over v: max v U 0 = n(1 v) ρ + λvl ρ 2 θ. The above maximand is a concave function in v and thus yieds a unique optima aocation of abor between research and production activities. This socia panner s aocation, denoted by v SP, is given by: v SP = λl ρθ λl. (17) This ies stricty in the interior of [0, 1] if λl > ρθ, namey if Assumption 1 hods. Moreover, comparing this with the aocation that woud emerge in the market equiibrium from (11), one immediatey has the foowing resut: Proposition 1 The share of abor that a socia panner woud aocate to research is stricty arger than that which is observed in the market equiibrium, namey v SP > v eqm. The decentraized equiibrium in our mode therefore unambiguousy yieds ess investment in R&D effort reative to the sociay-optima eve. One can moreover see that the reative extent to which v eqm fas short of v SP, namey (v SP v eqm )/v SP is increasing in θ. Intuitivey, the ess fat-taied is the Pareto distribution of productivity improvement draws, the ess attractive are the potentia private 14

18 returns (profits) from R&D, and hence the greater the extent of under-investment in R&D in the market equiibrium reative to the socia optimum. The iterature on endogenous growth in the presence of knowedge spiovers has highighted severa externaities that drive a wedge between the market and socia-panner outcomes (e.g., Grossman and Hepman, 1991; Aghion and Howitt, 1992), and these forces are present too in our mode. First, there is an intertempora spiover effect arising from the cumuative nature of innovation: Firms appy a higher effective discount rate when evauating the vaue of a patent because they do not internaize the positive knowedge spiovers from their innovation on future productivity improvements. Second, there is an appropriabiity effect, in that the private profits which firms earn are in genera smaer than the fu gains to consumer surpus that each innovation generates. Third, a business-steaing effect is at pay, since innovation effort erodes the profits of preceding innovators in a way that a socia-panner woud want to fuy internaize. The first two of these effects tend to decrease R&D in the market equiibrium reative to the panner s probem, whie the ast effect pushes firms toward over-investing in R&D. Proposition 1 impies that in our mode, the former two effects must dominate the atter businesssteaing mechanism. 19 We can in fact make a more precise statement concerning the reative importance of these three externaities. By rearranging (11), observe that the market aocation of abor to research activity is determined as the soution to: λ(1 v)l (ρ+λvl)θ panner s probem impies that v SP soves: λ(1 v)l ρθ = 1. On the other hand, the first-order condition of the socia = 1. Thus, the ony wedge between the two soutions arises from the different discount rates that are respectivey appied: In the market equiibrium, firms use a higher discount rate of ρ + λvl, which takes into account the fow probabiity of suffering a compete profit oss to a new innovation, on top of the socia discount rate. The ony externaity that is reevant in our mode is thus the intertempora spiover effect; evidenty, the appropriabiity and business-steaing effects must offset each other exacty. 3 Inventive Step Poicy The under-investment in R&D activity which our baseine mode features gives rise to the possibiity of wefare-improving poicy interventions. We turn next to anayze a poicy instrument that can potentiay achieve this, namey a minimum inventive step requirement for an idea to be patentabe and non-infringing. As argued in the Introduction, this captures a key dimension of the non-obviousness criterion commony stipuated in patent codes. In order to focus attention on this NIS requirement, we 19 Note that the presence of monopoy-pricing power per se does not distort abor aocations in our mode. The reason is that a firms charge the same markup in expectation (drawn from the standardized Pareto distribution, µ(m)), so that the aocation of production abor across varieties cannot be improved upon ex ante. See the reated discussion in Grossman and Hepman (1991), p

19 maintain the assumption that patents do not expire; we sha return to incorporate considerations reated to a finite patent ength ater in Section Poicy setup: The patenting environment We retain the cumuative innovation process described in Section 2.1, where each successive productivity improvement is an independent draw from the standardized Pareto distribution. Even though patenting was necessary in the baseine mode to confer a successfu innovator with short-term monopoy power, the patent authority there payed a reativey passive roe as a arriving ideas woud automaticay improve upon the frontier technoogy and hence quaify for a patent. Suppose now that the government announces at date τ = 0 that there wi be a non-infringing inventive step (NIS) requirement equa to B 1 with immediate effect: A new patent wi be granted if and ony if the k-th idea to arrive for a given variety improves upon the productivity of the (k 1)-th idea by at east B. More formay, this k-th idea is not eigibe for a patent if Z (k) [Z (k 1), BZ (k 1) ), an event that occurs with probabiity 1 B θ, based on the Pareto distribution in (1). In this situation, the firm in possession of this new idea woud have no incentive to produce the good even though it embodies an incrementay better technoogy, given that it woud have no ega right to market the good. Consequenty, non-patentabe ideas, which aso infringe, are not marketed; the underying knowedge does not spread to the rest of society and hence cannot be buit upon by subsequent innovators. This serves to capture the idea that the pubic discosure of technica knowhow that each patent appication requires is a crucia patform for faciitating knowedge diffusion. We make expicit two remarks on this treatment of non-patentabe ideas. First, note that the owner of a non-patentabe idea woud hersef be unabe to buid cumuativey on it, since future research effort cannot be directed toward a particuar variety in this setup. Second, we rue out the possibiity of this owner seing the non-patentabe idea to the incumbent patent-hoder of the variety. Any such attempted sae woud require the owner of the idea to first share information about it with the patent-hoder (for exampe, to show proof of concept). However, upon discosure, the incumbent patent-hoder woud then be abe to appropriate the knowedge without compensating the owner of the idea, and further use it for market production with no fear of ega action since the idea coud not be patented in the first pace. The owner of the idea woud thus have no incentive to attempt such a transaction. On the other hand, if the k-th idea to arrive were to satisfy Z (k) [BZ (k 1), ), an event that occurs with probabiity B θ, the firm in possession of this idea woud indeed patent it in order to subsequenty enjoy the profits from marketing the good. Since what is important now is whether an idea is patentabe or not, we et Z (k) denote the random variabe associated with the productivity eve of the k-th patentabe idea to arrive after time τ = 0. Moreover, et ζ (k) = Z (k) / Z (k 1) denote the productivity improvement 16